This application of inverse eigenvalues, conditioned on the presence of the corresponding eigenvectors, is the basic function of HHL.
By using quantum phase estimation (QPE) [1]}, it is possible to extract and apply these eigenvalue factors in an efficient manner.
However, this phase estimation requires the implementation of a problem-dependent unitary in terms of basic quantum gates.
Generally the unitary \(e^{iA}\)  is employed for this purpose, but the process of decomposing this unitary into a sequence of basic quantum gates is nontrivial.
Therefore the process of implementing this operation on real quantum hardware presents a bottleneck for the procedure.
In this work we consider a particular method, inspired by research into quantum walks, which produces a unitary with a known decomposition into basic gates.
